The expression \( \sqrt{25 - x^2} \) is a mathematical function that describes the upper half of a circle with a radius of 5 centered at the origin \((0, 0)\).

### Key Features of the Function:

1. **Domain**:
   - The expression inside the square root must be non-negative because the square root of a negative number is not a real number.
   - \( 25 - x^2 \geq 0 \) implies \( x^2 \leq 25 \), which means \( -5 \leq x \leq 5 \).
   - So, the domain of the function is \( [-5, 5] \).

2. **Range**:
   - Since \( \sqrt{25 - x^2} \) represents the upper half of a circle, the maximum value of the square root occurs when \( x = 0 \).
   - When \( x = 0 \), \( \sqrt{25 - 0^2} = \sqrt{25} = 5 \).
   - The smallest value occurs when \( x = \pm 5 \), giving \( \sqrt{25 - 25} = \sqrt{0} = 0 \).
   - Thus, the range of the function is \( [0, 5] \).

3. **Graph**:
   - The function \( y = \sqrt{25 - x^2} \) describes the upper semicircle of a circle centered at the origin with a radius of 5.
   - The graph is a semicircle extending from \( x = -5 \) to \( x = 5 \) and from \( y = 0 \) to \( y = 5 \).

4. **Symmetry**:
   - The function is symmetric about the y-axis because if \( x \) is replaced with \( -x \), the expression remains the same.

5. **Geometric Interpretation**:
   - The full equation \( x^2 + y^2 = 25 \) represents a circle with a radius of 5 centered at the origin.
   - The given expression corresponds to the positive (upper) part of this circle.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:
1. What is the equation for the lower half of the circle with radius 5 centered at the origin?
2. How would you describe the function \( \sqrt{r^2 - x^2} \) for a general radius \( r \)?
3. How does the domain change if the equation is \( \sqrt{16 - x^2} \)?
4. Can you find the area of the semicircle represented by \( y = \sqrt{25 - x^2} \)?
5. What happens to the graph if the expression is \( y = -\sqrt{25 - x^2} \)?

**Tip:** When dealing with square root functions like \( \sqrt{25 - x^2} \), always consider the domain and range carefully, as the function is only defined for values where the expression inside the square root is non-negative.

Understand the mathematical function \( \sqrt{25 - x^2} \) describing the upper half of a circle with a radius of 5 centered at the origin. Discover its domain, range, graph, symmetry, and geometric interpretation. Suitable for Grades 7-9.

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